Presentations of Categories of Modules using the Cautis-Kamnitzer-Morrison Principle

TL;DR

This paper generalizes a categorical approach to modules over Lie algebras and algebras, providing presentations of categories of modules via duality theorems and diagrammatic methods, with applications to Schur-Weyl and Brauer-Schur-Weyl dualities.

Contribution

It extends the Cautis-Kamnitzer-Morrison principle to derive presentations of module categories using duality theorems, connecting them to quotients of Lusztig's idempotented form.

Findings

Categorical presentations of modules using duality theorems.

Diagrammatic descriptions of subcategories in representation categories.

Explicit tensor product descriptions for permutation modules.

Abstract

We use duality theorems to obtain presentations of some categories of modules. To derive these presentations we generalize a result of Cautis-Kamnitzer-Morrison [arXiv:1210.6437v4]: Let $\mathfrak{g}$ be a reductive Lie algebra, and $A$ an algebra, both over $\mathbb{C}$. Consider a $(\mathfrak{g} , A)$-bimodule $P$ in which (a) $P$ has a multiplicity free decomposition into irreducible $(\mathfrak{g} , A)$-bimodules. (b) $P$ is "saturated" i.e. for any irreducible $\mathfrak{g}$-module $V$, if every weight of $V$ is a weight of $P$, then $V$ is a submodule of $P$. We show that statements (a) and (b) are necessary and sufficient conditions for the existence of an isomorphism of categories between the full subcategory of $\mathcal{R}ep A$ whose objects are $\mathfrak{g}$-weight spaces of $P$, and a quotient of the category version of Lusztig's idempotented form, $\dot{{\mathcal{U}}} \mathfrak{g}$, formed by setting to zero all morphisms factoring through a collection of objects in $\dot{{\mathcal{U}}} \mathfrak{g}$ depending on $P$. This is essentially a categorical version of the identification of generalized Schur algebras with quotients of Lusztig's idempotented forms given by Doty in [arXiv:math/0305208]. Applied to Schur-Weyl Duality we obtain a diagrammatic presentation of the full subcategory of $\mathcal{R}ep S_d$ whose objects are direct sums of permutation modules, as well as an explicit description of the $\otimes$-product of morphisms between permutation modules. Applied to Brauer-Schur-Weyl Duality we obtain diagrammatic presentations of subcategories of $\mathcal{R}ep \mathcal{B}_{d}^{(- 2n)}$ and $\mathcal{R}ep \mathcal{B}_{r,s}^{(n)}$ whose Karoubi completion is the whole of $\mathcal{R}ep \mathcal{B}_{d}^{(- 2n)}$ and $\mathcal{R}ep \mathcal{B}_{r,s}^{(n)}$ respectively.